Counting tropical morphisms from metric graphs to metric trees

We consider a tropical analogue to the following question: given a smooth complex algebraic curve X of even genus 2g', how many isomorphism classes of non-constant rational maps f : X → ℂℙ¹ with deg f = g' + 1 are there? In this talk we introduce the relevant tropical objects and sketch a proof that, with the appropriate multiplicity, the tropical count coincides with the classical count. The main idea is to organize both the tropical maps and the tropical curves into moduli spaces, and prove that the space of maps covers the space of curves. The desired number is then the degree of this branched covering. By looking at a particular fiber in the covering, a fiber related to chains of loops, this count is calculated to be a number that appears frequently in combinatorics.